Notes on Chapters 3 and 4 of Dedekind’s Theory of Algebraic Integers

These notes are intended as a high-level overview of some of the central ideas of Dedekind’s theory of ideals, as presented in Chapters 3 and 4. We saw at the end of Chapter 2 (and in the last set of notes) that Dedekind’s goal is to extend the unique factorization of ideals in Z[ √−5] to the unique factorization of ideals in the ring of integers of an arbitrary number field, with “proofs based immediately on fundamental characteristics, rather than on calculation.” I find it useful to recast Dedekind’s example in Chapter 2 in three components: • First, introducing the ring of “integers,” Z[√−5], as the set {a + b √−5 | a, b ∈ Z}. • Second, using calculations and a particular representation of the ideals in Z[ √−5] to derive properties of these ideals. • Third, using these results (and additional calculations) to prove unique factorizations for the ring of ideals.